Subgroups — #13054266

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#66631	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66577	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#66575	C₂	1	1	yes	maximal, characteristic
6	C₃×C₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620529	C₂	1	1	yes	maximal, characteristic, complemented
8	#89761	—	3	1	no	maximal, Hall
9	C₂³×D₂₁	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
12	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×D₂₁⋊C₄	—	2	2	no	—
17	C₂×D₂₁⋊C₄	—	2	2	no	—
18	C₂×D₂₁⋊C₄	—	2	2	no	—
19	#66631	C₄	1	4	yes	complemented
20	C₇×C₄⋊Dic₃	—	2	2	no	—
21	#66577	—	2	2	no	—
22	#66575	—	2	2	no	—
23	C₂²×C₄×D₇	—	3	1	no	—
24	C₂×C₇×C₄⋊C₄	—	3	1	no	—
25	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
26	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
27	C₂×Dic₇⋊C₄	—	3	1	no	—
28	C₇⋊#95	—	3	1	no	—
29	C₇⋊#95	D₃	1	1	yes	characteristic, complemented
30	#4668	—	7	1	no	maximal, Hall
31	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
32	C₂²×D₂₁	—	2	1	no	—
33	C₂²×D₂₁	—	2	1	no	—
34	C₂²×D₂₁	—	2	1	no	—
35	C₂²×D₂₁	—	2	1	no	—
36	C₂²×D₂₁	C₂×C₄	1	2	yes	—
37	C₂²×D₂₁	D₄	1	2	yes	—
38	C₂²×D₂₁	Q₈	1	2	yes	—
39	C₂×C₃×C₄×C₇	—	2	1	no	—
40	C₂×C₃×C₄×C₇	—	2	1	no	—
41	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	complemented
42	C₂×Dic₂₁	—	2	1	no	—
43	C₂×Dic₂₁	—	2	1	no	—
44	C₂×Dic₂₁	C₂×C₄	1	2	yes	complemented
45	C₂×C₇×Dic₃	—	2	1	no	—
46	C₂×C₇×Dic₃	—	2	1	no	—
47	C₂×C₇×Dic₃	—	2	1	no	—
48	C₂×C₇×Dic₃	—	2	1	no	—
49	C₂×C₇×Dic₃	—	2	1	no	—
50	C₂×C₇×Dic₃	D₄	1	2	yes	—
51	C₂×C₃×Dic₇	—	2	1	no	—
52	C₂×C₃×Dic₇	—	2	1	no	—
53	C₂×C₃×Dic₇	—	2	1	no	—
54	C₂×C₃×Dic₇	—	2	1	no	—
55	C₂×C₃×Dic₇	—	2	1	no	—
56	C₂×C₃×Dic₇	D₄	1	2	yes	—
57	D₂₁⋊C₄	—	2	8	no	—
58	C₂³×D₇	—	3	1	no	—
59	C₂²×C₄×C₇	—	3	1	no	—
60	C₂²×C₄×C₇	—	3	1	no	—
61	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
62	C₂²×Dic₇	—	3	1	no	—
63	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
64	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×D₇	—	6	2	no	—
66	C₂×C₄×D₇	—	6	2	no	—
67	C₂×C₄×D₇	—	6	2	no	—
68	C₇×C₄⋊C₄	—	6	2	no	—
69	(C₂×D₇)⋊C₄	—	3	4	no	—
70	(C₂×D₇)⋊C₄	—	6	2	no	—
71	Dic₇⋊C₄	—	6	2	no	—
72	C₂²×C₄×D₃	—	7	1	no	—
73	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
74	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
75	C₂×C₄⋊Dic₃	D₇	1	1	yes	characteristic, complemented
76	C₂×Dic₃⋊C₄	—	7	1	no	—
77	C₃×#95	—	7	1	no	—
78	C₃⋊#95	—	7	1	no	—
79	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₃×C₇	#41	1	1	yes	characteristic
85	C₂²×C₃×C₇	#41	1	1	yes	characteristic
86	C₂×D₂₁	—	2	2	no	—
87	C₂×D₂₁	—	2	2	no	—
88	C₂×D₂₁	—	2	2	no	—
89	C₂×D₂₁	—	2	2	no	—
90	C₂×D₂₁	—	2	2	no	—
91	C₂×D₂₁	—	2	2	no	—
92	C₂×D₂₁	C₄⋊C₄	1	4	yes	complemented
93	C₃×C₄×C₇	—	2	2	no	—
94	Dic₂₁	—	2	2	no	—
95	C₃×Dic₇	—	2	2	no	—
96	C₃×Dic₇	—	4	1	no	—
97	C₇×Dic₃	—	2	2	no	—
98	C₇×Dic₃	—	4	1	no	—
99	#385	—	21	1	no	Sylow
100	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
101	C₂²×D₇	—	3	2	no	—
102	C₂²×D₇	—	3	2	no	—
103	C₂²×D₇	—	3	2	no	—
104	C₂²×D₇	—	6	1	no	—
105	C₂²×D₇	—	6	1	no	—
106	C₂²×D₇	—	6	1	no	—
107	C₂²×D₇	—	6	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	3	2	no	—
111	C₂×C₄×C₇	—	6	1	no	—
112	C₂×C₄×C₇	—	6	1	no	—
113	C₂×C₄×C₇	—	6	1	no	—
114	C₂×C₄×C₇	—	6	1	no	—
115	C₂×C₄×C₇	—	6	1	no	—
116	C₂×C₄×C₇	C₄×D₃	1	2	yes	complemented
117	C₂×Dic₇	—	2	1	no	—
118	C₂×Dic₇	—	2	1	no	—
119	C₂×Dic₇	—	2	1	no	—
120	C₂×Dic₇	—	2	1	no	—
121	C₂×Dic₇	—	2	1	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	6	1	no	—
124	C₂×Dic₇	—	6	1	no	—
125	C₂×Dic₇	D₁₂	1	2	yes	—
126	C₄×D₇	—	6	8	no	—
127	C₂³×D₃	—	7	1	no	—
128	C₂²×C₃×C₄	—	7	1	no	—
129	C₂²×C₃×C₄	—	7	1	no	—
130	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
131	C₂²×Dic₃	—	7	1	no	—
132	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
133	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
134	C₂×C₄×D₃	—	14	2	no	—
135	C₂×C₄×D₃	—	14	2	no	—
136	C₂×C₄×D₃	—	14	2	no	—
137	(C₂×D₃)⋊C₄	—	7	4	no	—
138	(C₂×D₃)⋊C₄	—	14	2	no	—
139	C₄⋊Dic₃	—	2	2	no	—
140	Dic₃⋊C₄	—	14	2	no	—
141	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₃×C₇	#120	1	1	yes	characteristic
145	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₃×C₇	#123	1	1	yes	characteristic
148	D₂₁	—	2	4	no	—
149	C₂³×C₄	—	21	1	no	—
150	C₂×C₄⋊C₄	—	3	1	no	—
151	C₂×C₄⋊C₄	—	21	1	no	—
152	C₂×C₂²⋊C₄	—	21	1	no	—
153	C₂×C₂²⋊C₄	—	21	1	no	—
154	#95	—	7	1	no	—
155	#95	—	21	1	no	—
156	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
158	C₂²×C₇	#235	1	1	yes	characteristic
159	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
160	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
161	C₂²×C₇	#237	1	1	yes	characteristic
162	C₂²×C₇	D₃×Q₈	1	1	yes	characteristic
163	C₂×D₇	—	3	4	no	—
164	C₂×D₇	—	6	2	no	—
165	C₂×D₇	—	6	2	no	—
166	C₂×D₇	—	6	2	no	—
167	C₂×D₇	—	6	2	no	—
168	C₂×D₇	—	6	2	no	—
169	C₂×D₇	—	6	2	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	6	2	no	—
172	C₄×C₇	—	12	1	no	—
173	Dic₇	—	2	2	no	—
174	Dic₇	—	4	1	no	—
175	Dic₇	—	6	2	no	—
176	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
177	C₂²×D₃	—	7	2	no	—
178	C₂²×D₃	—	7	2	no	—
179	C₂²×D₃	—	7	2	no	—
180	C₂²×D₃	—	14	1	no	—
181	C₂²×D₃	—	14	1	no	—
182	C₂²×D₃	—	14	1	no	—
183	C₂²×D₃	—	14	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	7	2	no	—
187	C₂×C₃×C₄	—	14	1	no	—
188	C₂×C₃×C₄	—	14	1	no	—
189	C₂×C₃×C₄	—	14	1	no	—
190	C₂×C₃×C₄	—	14	1	no	—
191	C₂×C₃×C₄	—	14	1	no	—
192	C₂×C₃×C₄	C₄×D₇	1	2	yes	complemented
193	C₂×Dic₃	—	2	1	no	—
194	C₂×Dic₃	—	2	1	no	—
195	C₂×Dic₃	—	2	1	no	—
196	C₂×Dic₃	—	2	1	no	—
197	C₂×Dic₃	—	2	1	no	—
198	C₂×Dic₃	—	7	2	no	—
199	C₂×Dic₃	—	14	1	no	—
200	C₂×Dic₃	—	14	1	no	—
201	C₂×Dic₃	C₂²⋊D₇	1	2	yes	—
202	C₄×D₃	—	14	8	no	—
203	C₃×C₇	#385	1	1	yes	characteristic, complemented, Hall
204	C₂⁴	—	21	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	7	1	no	—
208	C₂²×C₄	—	7	1	no	—
209	C₂²×C₄	—	21	1	no	—
210	C₂²×C₄	—	42	2	no	—
211	C₂²×C₄	—	42	2	no	—
212	C₂²×C₄	—	42	2	no	—
213	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
214	C₄⋊C₄	—	6	2	no	—
215	C₄⋊C₄	—	42	2	no	—
216	C₂²⋊C₄	—	21	4	no	—
217	C₂²⋊C₄	—	42	2	no	—
218	C₂×C₇	C₄×D₁₂	1	1	yes	characteristic
219	C₂×C₇	#67⋊C₄	1	1	yes	characteristic
220	C₂×C₇	#882	1	1	yes	characteristic
221	C₂×C₇	#883	1	1	yes	characteristic
222	C₂×C₇	D₃×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₇	#896	1	1	yes	characteristic
224	C₂×C₇	#897	1	1	yes	characteristic
225	D₇	—	6	4	no	—
226	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
227	C₂²×C₃	#1157	1	1	yes	characteristic
228	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
229	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
230	C₂²×C₃	Q₈×D₇	1	1	yes	characteristic
231	C₂²×C₃	#1161	1	1	yes	characteristic
232	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
233	C₂×D₃	—	7	4	no	—
234	C₂×D₃	—	14	2	no	—
235	C₂×D₃	—	14	2	no	—
236	C₂×D₃	—	14	2	no	—
237	C₂×D₃	—	14	2	no	—
238	C₂×D₃	—	14	2	no	—
239	C₂×D₃	—	14	2	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	14	2	no	—
242	C₃×C₄	—	28	1	no	—
243	Dic₃	—	2	2	no	—
244	Dic₃	—	4	1	no	—
245	Dic₃	—	14	2	no	—
246	C₂³	—	21	2	no	—
247	C₂³	—	21	2	no	—
248	C₂³	—	21	2	no	—
249	C₂³	—	42	1	no	—
250	C₂³	—	42	1	no	—
251	C₂³	—	42	1	no	—
252	C₂³	—	42	1	no	—
253	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	6	1	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	14	1	no	—
264	C₂×C₄	—	14	1	no	—
265	C₂×C₄	—	14	1	no	—
266	C₂×C₄	—	14	1	no	—
267	C₂×C₄	—	14	1	no	—
268	C₂×C₄	—	21	2	no	—
269	C₂×C₄	—	42	8	no	—
270	C₂×C₄	—	42	1	no	—
271	C₂×C₄	—	42	1	no	—
272	C₂×C₄	D₂₁⋊C₄	1	2	yes	complemented
273	C₇	#4668	1	1	yes	characteristic, complemented, Sylow
274	C₂×C₃	D₇×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₃	#6456	1	1	yes	characteristic
276	C₂×C₃	#6457	1	1	yes	characteristic
277	C₂×C₃	#6459	1	1	yes	characteristic
278	C₂×C₃	C₄×C₂²⋊D₇	1	1	yes	characteristic
279	C₂×C₃	#6500	1	1	yes	characteristic
280	C₂×C₃	#6509	1	1	yes	characteristic
281	D₃	—	14	4	no	—
282	C₂²	—	21	4	no	—
283	C₂²	—	42	2	no	—
284	C₂²	—	42	2	no	—
285	C₂²	—	42	2	no	—
286	C₂²	—	42	2	no	—
287	C₂²	—	42	2	no	—
288	C₂²	—	42	2	no	—
289	C₂²	D₂₁⋊Q₈	1	1	yes	characteristic
290	C₂²	#66677	1	1	yes	characteristic
291	C₂²	D₇×D₁₂	1	1	yes	characteristic
292	C₂²	#66681	1	1	yes	characteristic
293	C₂²	#66683	1	1	yes	characteristic
294	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
295	C₂²	D₃×C₂²⋊D₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	6	2	no	—
298	C₄	—	12	1	no	—
299	C₄	—	14	2	no	—
300	C₄	—	28	1	no	—
301	C₄	—	42	2	no	—
302	C₃	#89761	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	42	4	no	—
304	C₂	#10620918	1	1	yes	characteristic
305	C₂	#10620980	1	1	yes	characteristic
306	C₂	#10620981	1	1	yes	characteristic
307	C₂	#10620986	1	1	yes	characteristic
308	C₂	#10620999	1	1	yes	characteristic
309	C₂	#10621037	1	1	yes	characteristic
310	C₂	#10621118	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₃×D₇
commutator	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle, cosocle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₂₁	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#385
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₇×C₄⋊Dic₃	C₂	semidirect
4. C₂×#66577	C₂	semidirect
6. C₃×C₇⋊#95	C₂	semidirect
7. #10620529	C₂	semidirect
19. #66631	C₄	semidirect
29. C₇⋊#95	D₃	semidirect
41. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
44. C₂×Dic₂₁	C₂×C₄	semidirect
75. C₂×C₄⋊Dic₃	D₇	semidirect
92. C₂×D₂₁	C₄⋊C₄	semidirect
116. C₂×C₄×C₇	C₄×D₃	semidirect
192. C₂×C₃×C₄	C₄×D₇	semidirect
203. C₃×C₇	#385	semidirect
272. C₂×C₄	D₂₁⋊C₄	semidirect
273. C₇	#4668	semidirect
302. C₃	#89761	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₂₁⋊C₄	2
C₂×#66631	2
C₂×C₇×C₄⋊Dic₃	2
C₂×#66577	2
C₂×#66575	2
C₃×C₇⋊#95	2
#10620529	2
#89761	3
#4668	7